ABSTRACT
Recently, odd-even high-order harmonic generation (HHG) emitted from asymmetric
molecules has paid much attention since its application in extracting molecular structures and
probing molecular dynamics. In which, the ratio of the even and odd harmonics in HHG spectra is
an important quantity characterized the molecular asymmetry. This even-to-odd ratio has been
explained by the gerade and ungerade components of the temporal transition dipole. Though, the
separating procedure of the temporal transition dipole has not been published in the previous
study. In this paper, we present the detailed procedure of separating the gerade and ungerade
components of the temporal transition dipole using the Floquet theorem. Besides, we explain the
correlation between the intensities of even and odd harmonics by the new approach, using the
induced dipole acceleration.

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TẠP CHÍ KHOA HỌC
TRƯỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINH
Tập 17, Số 3 (2020): 433-444
HO CHI MINH CITY UNIVERSITY OF EDUCATION
JOURNAL OF SCIENCE
Vol. 17, No. 3 (2020): 433-444
ISSN:
1859-3100 Website:
Research Article*
SEPARATING GERADE AND UNGERADE COMPONENTS
OF TEMPORAL TRANSITION DIPOLE BY FLOQUET THEOREM
Phan Thi Ngoc Loan
Ho Chi Minh City University of Education
Corresponding author: Phan Thi Ngoc Loan – Email: loanptn@hcmue.edu.vn
Received: February 10, 2020; Revised: March 19, 2020; Accepted: March 23, 2020
ABSTRACT
Recently, odd-even high-order harmonic generation (HHG) emitted from asymmetric
molecules has paid much attention since its application in extracting molecular structures and
probing molecular dynamics. In which, the ratio of the even and odd harmonics in HHG spectra is
an important quantity characterized the molecular asymmetry. This even-to-odd ratio has been
explained by the gerade and ungerade components of the temporal transition dipole. Though, the
separating procedure of the temporal transition dipole has not been published in the previous
study. In this paper, we present the detailed procedure of separating the gerade and ungerade
components of the temporal transition dipole using the Floquet theorem. Besides, we explain the
correlation between the intensities of even and odd harmonics by the new approach, using the
induced dipole acceleration.
Keywords: HHG; odd-even; symmetry; transition dipole; dipole acceleration
1. Introduction
In recent decades, laser-matter interaction is an interesting topic attracting much
attention of the community studying on the strong field physics and attosecond science
(Ghimire, & Reis, 2019; Lewenstein, Balcou, Ivanov, L’Huillier, & Corkum, 1994; Li et
al., 2017; McPherson et al., 1987). When atoms, molecules are exposed to an intense laser,
one of the nonlinear effects is the emission of photons, whose frequency is many times
greater than the irradiated laser’s one, and called the high-order harmonic generation
(HHG) (Lewenstein et al., 1994). The HHG can be applied to extract the atomic/molecular
structure (Itatani et al., 2004; Lein, Hay, Velotta, Marangos, & Knight, 2002), or probing
the ultrafast dynamics in femtosecond or attosecond scale in molecules (Borot et al., 2012).
The HHGs from multi-electron atoms or molecules are popularly observed in
laboratories (Itatani et al., 2004; McPherson et al., 1987). However, the theoretical
simulation for the multi-electron system is difficult due to the limitation of the computing
Cite this article as: Phan Thi Ngoc Loan (2020). Separating gerade and ungerade components of temporal
transition dipole by Floquet theorem. Ho Chi Minh City University of Education Journal of Science, 17(3),
433-444.
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HCMUE Journal of Science Vol. 17, No. 3 (2020): 433-444
resources (Abu-Samha, & Madsen, 2010; Lein et al., 2002; Bin Zhang, Yuan, & Zhao,
2014). To overcome this difficulty, the approximation models such as the strong-field
approximation (SFA) (Lewenstein et al., 1994; Zhou, Tong, Zhao, & Lin, 2005),
quantitative rescattering model (Le, Lucchese, Morishita, & Lin, 2009), or quantum-orbit
theory (Milošević, & Becker, 2002). Besides the approximate methods, another method is
ab initio calculating the HHG from the multi-electron system by using the single-active
electron (SAE) model (Abu-Samha, & Madsen, 2010). Accordingly, only one electron
from the high-occupied molecular orbital (HOMO) interacts with an intense laser.
Meanwhile, the remaining electrons are attached to the nuclei and called core-electron. The
effect caused by the core-electron is called the dynamic core-electron polarization (DCeP).
The DCeP significantly affects the destructive minimum in the HHG spectra emitted
from CO (Le, Hoang, Tran, & Le, 2018; Bin Zhang et al., 2014) or CO2 (Le, Vu, Ngo, &
Le, 2019) molecules. Besides, DCeP effects on the HHG intensity near the cutoff of the
HHG from CO (Le, Hoang, Tran, & Le, 2018; Bin Zhang et al., 2014) và CO2 (Le, Vu,
Ngo, & Le, 2019). Recently, we have also indicated the influence of the DCeP on the
correlation between the intensity of the even and odd harmonics in the HHG of CO and
NO molecules (Le, & Phan, 2020; Phan, Le, Hoang, & Le, 2019). To interpret this
phenomenon, we have calculated the gerade and ungerade components of the temporal
transition dipole (Le, & Phan, 2020; Phan et al., 2019). Accordingly, while the odd
harmonics are caused by the recombination of electrons into the gerade part, the even
harmonics – into the ungerade part of HOMO of CO molecule at the recombination time. It
is noted that the wavefunction of an asymmetric molecule, like CO, does not have a certain
parity. Therefore, we have to separate the Floquet wavefunction into two components –
gerade and ungerade ones based on the Floquet theory. It should be emphasized that the
Floquet wavefunction is different from the time-dependent wavefunction by a phase factor.
Meanwhile, the quantity is calculated by the numerical method is the wavefunction, not
Floquet wavefunction. These facts lead to the dificulty in wavefunction separation. In
(Phan et al., 2019), we have succesfully separated the gerade and ungerade components of
the Floquet wavefunction, and the corresponding temporal transition dipole of CO
molecule, however, the separating procedure is not published.
Besides, in the works (Hu, Li, Liu, Li, & Xu, 2017), Hu et al. have claimed that
when a CO molecule is subjected to a laser whose polarization vector is perpendicular to
the molecular axis, the HHG component parallel to the electric vector contains only odd
harmonics. On the contrary, the HHG component whose polarization perpendicular to the
laser electric-vector consists of only even harmonics. The authors have explained that the
pure odd and pure even HHG spectra based on the gerade and ungerade of the induced
dipole acceleration. So the question is that, besides the approach of using temporal
transition dipole as in (Le, & Phan, 2020; Phan et al., 2019), is it possible to use induced
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HCMUE Journal of Science Phan Thi Ngoc Loan
dipole acceleration to explain the correlation between the intensity of odd and even
harmonics in HHG?
In this paper, we present the detailed procedure for separation of the gerade and
ungerade temporal transition dipole when electron recombines into the HOMO of CO
molecule based on the Floquet theory. This explicit representation is meaningful in the
explanation of the property of HHG spectra from asymmetric molecules. After that, we
interpret the correlation between the intensity of the even and odd harmonics from CO
molecules when considering and neglecting the DCeP effect by using the induced dipole
acceleration. To calculate the HHG spectra, we employ the method of solving the
Schrödinger equation (TDSE) in the framework of SAE approximation.
2. Theoretical background
In this section, we briefly present the model of CO molecule interacting with the
laser, and the TDSE method to obtain the HHG spectra. Then, we recall the basis of the
theory of strong-field approximation and the calculation method of the transition dipole.
Finally, the Floquet theorem for the temporal-spatial periodic system is presented.
2.1. Molecular model and TDSE method
Hamiltonian of the laser-molecular system in the atomic unit has the following form
0
ˆ ˆ ˆ( ) ( , ),iH H H t= +r r (1)
where
2
0
ˆ ( ) ( )
2 SAE
H V∇= − +r r , and ( )SAEV r is the molecular potential constructed by the
SAE model (see details in (C.-T. Le et al., 2018; Phan et al., 2019)). The parameters are
chosen so that the HOMO energy of CO (5σ) reaches – 0.51 a.u.
The Hamiltonian characterized the laser-molecular coupling is separated into to
components
ˆ ( , ) ( , ) ( , )i L PH t V t V t= +r r r , (2)
where ( , ) ( )LV t t= ⋅r r E describes the interaction between the active electron and the laser
electric field; 3
ˆ( )( , ) cP
tV t
r
α
= −
E rr is the polarization potential considering the influence of
the core-electron polarization; ˆcα is the polarization tensor. The tensor values are
presented in (Phan et al., 2019).
The magnitude of the laser has the following form
( ) ( )sino oE t E f t tω= , (3)
where 0 0,E ω are the maximum amplitude and the carrier frequency of laser; ( )f t is the
laser envelope. In this study, we use a laser pulse with a duration of 10 optical cycles, and
the laser envelope has the trapezoidal form with one-cycle turn-on, one-cycle turn-off, and
eight cycles in the flat part. The laser intensity is 141.5 10× W/cm2. The polarization vector
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HCMUE Journal of Science Vol. 17, No. 3 (2020): 433-444
of the laser is aligned with the Oz axis. The CO model is present in Fig. 1. The molecular
axis makes with the Oz an angle θ – called alignment angle.
Figure 1. CO molecule in the molecular frame Ox’z’ and laboratory frame Oxz
To solve the Schrödinger equation for time-dependent Hamiltonian (1), we use the
method presented in (Le et al., 2018; Phan et al., 2019). After obtaining the time-dependent
wavefunction ( , )tψ r , the induced moment dipole parallel to the electric field is defined as
( ) ( , ) | | ( , ) .zD t t z tψ ψ= r r (4)
In this article, we present the results for the HHG with the parallel polarization. For
perpendicular HHG, the procedure is similar (Phan et al., 2019). The HHG spectra are
calculated as the square of the Fourier transform of the acceleration of the transition
moment (4). For solving TDSE, we use the computational parameters as presented in
(Phan et al., 2019).
2.2. Transition dipole
The TDSE method gives the result with high precision (Le et al., 2018, 2019; Phan et
al., 2019), while the approaches using the model provide the intuitive picture to explain the
physical mechanism. In which, the SFA model interprets the HHG by the three steps
(Lewenstein et al., 1994; Zhou et al., 2005). First, the electron tunnels into a continuum
then accelerates in the electric field; finally, when the electric field changes its directions,
the electron returns to the parent ion. The HHG is emitted at the last step.
According to SFA, the HHG intensity is proportional to the square of the transition
dipole from the continuum into the bound states, which characterizes the third step in the
three-step model (Itatani et al., 2004). Therefore, to clarify the feature and property of
HHG, the transition dipole is considered. For simplicity, the continuum wave functions are
assumed as plane waves. The transition dipole has the following form
( )( ) ( ) | | e ,izd z
ωω ψ ⋅= k rr (5)
where ω is the harmonic frequency; ( )ωk is the wavenumber satisfied the dispersion
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HCMUE Journal of Science Phan Thi Ngoc Loan
formula ( ) 2ω ω=k . In most works (Itatani et al., 2004; Bing Zhang, Chen, Jiang, &
Sun, 2013), the authors have claimed that the molecular HOMO is not deformed while
interacting with the laser; therefore, the wavefunction ( )ψ r is the initial wavefunction, i.e.,
when the molecule has not yet interacted with the laser.
However, to interpret the dynamic properties of a molecule, the temporal transition
dipole is used (Bin Zhang et al., 2014). Its expression is written as
( )( , ) ( , ) | | eizd t t z
ωω ψ ⋅= k rr . (6)
2.3. Floquet theorem
The Floquet theorem describes the property of the wavefunction of a temporal-spatial
periodic potential. This theorem has been presented in detail in many references (Bavli &
Metiu, 1993; Tal, Moiseyev, & Beswickf, 1993). Here, we summarise some main
equations.
Consider a system described by a Hamiltonian satisfied the periodic condition
0
ˆ ˆ( , ) ( , / 2),H t H t T= − +r r (7)
where 0T - an optical cycle. For this system, the time-dependent wavefunction obeys the
Floquet theorem, i.e., the wavefunction is presented as a product of an exponential factor
and Floquet wavefunction
( , ) ( , ),i tt e tεψ ϕ−=r r (8)
where ε is the Floquet quasi-energy, and ( , )tϕ r is the Floquet wavefunction satisfied the
symmetry condition
0( , ) ( , / 2).t t Tϕ ϕ= ± − +r r (9)
3. Results
In this section, first, we present the detailed procedure for separating the gerade and
ungerade components of temporal transition dipole. Then, this procedure is applied to
explain the correlation between the intensity of even and odd harmonics. Finally, we
interpret the ratio between the intensity of even and odd harmonics (referred to the even-to-
odd ratio) by the induced dipole acceleration.
3.1. Procedure for separating gerade and ungerade components of temporal transition
dipole
Usually, to investigate the intensity correlation between even and odd harmonics, the
gerade and ungerade components of transition dipole (5) are considered (Bing Zhang et al.,
2013). The work (Bing Zhang et al., 2013) has shown that the recombination of the
electron into the gerade wavefunction results in odd harmonic generation. On the contrary,
the recombination into the ungerade HOMO leads to the emission of the even harmonics.
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HCMUE Journal of Science Vol. 17, No. 3 (2020): 433-444
Figure 2. The HOMO of CO molecule in the molecular frame Ox’z’ at the instances 4.45T0 (a),
4.95T0 (b), and 5.45T0 (c) when neglecting the DCeP effect. The alignment angle is 45º
However, to investigate the influence of the DCeP on the HHG intensity correlation
between the even and odd orders, the temporal transition dipole (6) is needed since the
DCeP distorts the HOMO during the interaction (Phan et al., 2019; Bin Zhang et al., 2013),
results in changing the wavefunction parity. Therefore, we have to consider the temporal
wavefunction.
For atoms or symmetric molecules, the potential ( )SAEV r is inversion symmetry, i.e.,
their time-dependent Hamiltonian satisfies the symmetry condition (7). As a consequence,
the time-dependent wavefunction obeys the Floquet condition (8), (9) (Tal et al., 1993).
However, for asymmetric molecules, such as CO, the potential is not inversion symmetric
(see Fig. 2), that the Floquet wavefunction does not satisfy the symmetric condition (9),
i.e., it does not possess a certain parity. However, we can theoretically separate the
unsymmetric Floquet wave function into the symmetric ( , )g tϕ r and antisymmetric
( , )u tϕ r ones
( ) 0( , ) ( , / 2)( , )
2
g u t t Tt ϕ ϕϕ ± − += r rr . (10)
The time-dependent wavefunction also can be separated into two corresponding
components ( ) ( )( , ) ( , )g u i t g ut e tεψ ϕ−=r r . The corresponding temporal transition dipoles
( ) ( ) ( )( , ) ( , ) | | e .g u g u izd t t z
ωω ψ ⋅= k rr (11)
These components are responsible for generating the odd and even harmonics, respectively
(Phan et al., 2019).
However, in practice, the separating of the Floquet wavefunction (10) faces difficulty
since the solution from the TDSE is the time-dependent wavefunction, not Floquet
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HCMUE Journal of Science Phan Thi Ngoc Loan
wavefunction and Floquet quasi-energy. Fortunately, looking insight the Hamiltonian (1),
we realize that although it is not symmetric through the transformation
0: , / 2P t t T→− → +r r , but it is symmetric through 0' : ,P t t T→ → +r r . This fact can be
illustrated in Fig. 2 for HOMO of CO molecule at the instances 4.45T0, 4.95T0, và 5.45T0.
This HOMO is not symmetric through half-cycle translation but is repeated after each
optical cycle. Therefore, the Floquet wavefunction satisfies
0( , ) ( , )t t Tϕ ϕ= ± +r r , (12)
and the time-dependent wavefunction obeys
0
0( , ) ( , )
i Tt T e tεψ ψ+ = ±r r . (13)
By the Eq. (13) we can calculate the phase factor 0i Te ε from the time-dependent
wavefunction calculated by the TDSE method. After getting the phase factor, or Floquet
quasi-energy, the symmetric and antisymmetric components ( ) ( , )g u tψ r of the time-
dependent wavefunction is calculated as
0 0
0
( ) ( ) 0
/2 ( /2)
0
/2
0
( , ) ( , / 2)( , ) ( , )
2
( , ) ( , / 2)
2
( , ) ( , / 2) .
2
i t i t
g u i t g u
i T i t Ti t
i T
e t e t Tt e t
e t e e t T
t e t T
ε ε
ε
ε εε
ε
ϕ ϕ
ψ ϕ
ϕ ϕ
ψ ψ
− −
−
− +−
± − +
≡ =
± − +
=
± − +
=
r rr r
r r
r r
(14)
From these components, the gerade and ungerade temporal transition dipoles are
calculated by Eq. (11).
3.2. Application to explain the intensity correlation between even and odd harmonics
Figure 3. The parallel HHG emitted from CO molecule with alignment angle 45º
when including (SAE+P) and ignoring (SAE) DCeP effect
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HCMUE Journal of Science Vol. 17, No. 3 (2020): 433-444
Now we apply the above procedure to calculate the gerade and ungerade temporal
transition dipole to explain the intensity correlation between even and odd harmonics of
CO molecule. Figure 3 presents the component of HHG whose polarization is parallel to
the molecular axis for CO molecule at alignment angle 45º, when considering (SAE+P)
and neglecting (SAE) the DCeP effect. The results calculated by the TDSE method show
that the cutoff of the spectra is 29th order, is consistent with the predicted ones by the three-
step model (Lewenstein et al., 1994). Besides, the HHG spectra contain both odd and even
orders. Especially, for harmonics near the cutoff, when considering the DCeP, the intensity
of the even harmonics is lower than that of the odd ones. The intensities of the even
harmonics for both cases – including and neglecting the DCeP, are comparable.
Meanwhile, for the odd harmonics, its intensity for the case SAE is much less than that of
the SAE+P case. These results have been presented in our recent study (Phan et al., 2019).
To explain the intensity correlation between the even and odd harmonics near the
cutoff, we calculate the temporal transition dipole. First, we determine the recombination
time of electron resulted in harmonics near cutoff. Figure 4 shows the kinetic energy of the
returning electron. The results show that, for every half cycle, there is one instance that the
kinetic energy reached maximum - 3.17 pU , where pU is the ponderomotive energy. The
recombination instances are 0 0 0 01.45 ,1.95 ,2.45 ,2.95 ,...T T T T
Figure 4. The kinetic energy of the returning electron
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HCMUE Journal of Science Phan Thi Ngoc Loan
Figure 5. The transition dipole at 4.45T0 of the CO molecule with alignment angle 45º
Figure 5 presents the gerade and ungerade transition dipole at the recombination time
04.45T , corresponding to the harmonic 27
th at the cutoff according to the three-step model.
The results show that, when ignoring the DCeP effect (Fig. 5a), the ungerade transition
dipole is much greater than the gerade’s one, leading to the more efficient of even
harmonics compared to the odd’s one. Meanwhile, when including the DCeP (Fig. 5b), the
magnitude of the two components of transition dipole is comparable, leading to the
equivalence of odd and even harmonics. The correlation between the gerade and ungerade
transition dipoles plays a decisive role in the intensity correlation between the odd and
even harmonics in HHG spectra.
3.3. Explanation of intensity correlation between even and odd harmonics by induced
dipole acceleration
To continue, we explain the correlation between odd and even harmonics by the
approach differed from the one using in (Phan et al., 2019). Here, we use the induced
dipole acceleration calculated by the second differential of induced dipole moment written
by Eq.(4). Based on the nature of the Fourier transform, the odd harmonics in HHG is
caused by the ungerade induced dipole acceleration through the transformation
0 / 2t t T→ + . Meanwhile, the even harmonics are raised